scholarly journals On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions

2017 ◽  
Vol 184 (1) ◽  
pp. 77-103
Author(s):  
M. Cihat Dağlı ◽  
Mümün Can
Keyword(s):  
Eisenstein Series ◽  
Zeta Functions ◽  
Ramanujan’S Formula

scholarly journals Zeta functions on tori using contour integration

2015 ◽  
Vol 12 (02) ◽  
pp. 1550019
Author(s):  
Emilio Elizalde ◽  
Klaus Kirsten ◽  
Nicolas Robles ◽  
Floyd Williams
Keyword(s):  
Functional Equation ◽  
Eisenstein Series ◽  
Zeta Functions ◽  
Contour Integration ◽  
Alternative Proof ◽  
Functional Determinant ◽  
Dedekind Eta Function ◽  
Limit Formula ◽  
Complex Tori ◽  
The One

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla–Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well.

scholarly journals Around the Lipschitz Summation Formula

2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Wenbin Li ◽  
Hongyu Li ◽  
Jay Mehta
Keyword(s):  
Zeta Function ◽  
Eisenstein Series ◽  
Boundary Behavior ◽  
Summation Formula ◽  
Boundary Function ◽  
Essential Information ◽  
Long Run ◽  
Underlying Principle ◽  
Ramanujan’S Formula ◽  
Lerch Transcendent

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.

scholarly journals Multiple Dedekind zeta functions

2017 ◽  
Vol 2017 (722) ◽  
Author(s):  
Ivan Emilov Horozov
Keyword(s):  
Eisenstein Series ◽  
Zeta Functions ◽  
Quadratic Fields ◽  
Multiple Zeta Values ◽  
Imaginary Quadratic Fields ◽  
Iterated Integrals ◽  
Multiple Zeta ◽  
Zeta Values ◽  
New Type

AbstractIn this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler’s multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series [

scholarly journals Hecke’s Integral Formula for Relative Quadratic Extensions of Algebraic Number Fields

2008 ◽  
Vol 189 ◽  
pp. 139-154 ◽  
Author(s):  
Shuji Yamamoto
Keyword(s):  
Zeta Function ◽  
Eisenstein Series ◽  
Algebraic Number ◽  
Integral Formula ◽  
Zeta Functions ◽  
Quadratic Extension ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Limit Formula ◽  
Quadratic Extensions

AbstractLet K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker’s type which relates the 0-th Laurent coefficients at s = 1 of zeta functions of K and F.

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